nLab
Prüfer group

Contents

Definition

For a prime number pp, the Prüfer pp-group is defined uniquely up to isomorphism as the group where every element has exactly pp p thp^{th} roots. It is a divisible abelian group which can be described in several ways, for example:

As such, it is the initial algebra of the functor F:LCHAbLCHAbF: \mathbb{Z} \darr LCHAb \to \mathbb{Z} \darr LCHAb that pushes out along multiplication by p:p: \mathbb{Z} \to \mathbb{Z}.

Properties

The Prüfer pp-groups are the only infinite groups whose subgroups are totally ordered by inclusion. They are often useful as counterexamples in algebra; for example, a Prüfer group is an Artinian but not a Noetherian \mathbb{Z}-module.

Revised on August 27, 2014 03:55:54 by David Corfield (129.12.18.145)